the point lies on the ray (AR) going from the center of the circle (A) to the initial point (F)
Justification: Let AR (the radius of the circle of inversion) = 1
:where A is the center of the circle and R is on the circle
Because F is inside the circle, AF is less than 1, but greater than zero
We know: (AF') = (AR)(AR) / (AF)
so (AF') = (1) / (a number less than 1, but greater than zero)
therefore, (AF') is greater than 1, which puts the point F' ( the inverse of F)
the point lies on the ray going from the center of the circle to the initial point
Justification: Let AR (the radius of the circle of inversion) = 1
:where A is the center of the circle and R is on the circle
Becuase F is outside the circle, AF is greater than 1
We know: (AF') = (AR)(AR) / (AF)
so (AF') = (1) / (a number greater than 1)
therefore, (AF') is less than 1, which puts the point F' ( the inverse of F)
Justification: Let point F be on the circle of inversion
Let AR (the radius of the circle of inversion) = 1
:where A is the center of the circle and R is on the circle
Because F is on the circle AF = AR = 1
We know: (AF') = (AR)(AR) / (AF)
so (AF') = (1) / (1) = 1
therefore, F' ( the inverse of F) must also lie on the circle of inversion
The inverse of the center of the circle is at infinity.
Justification: Let Point F be at point A ( the center of the circle of inversion0
Let AR (the radius of the circle of inversion) = 1
: where A is the center of the circle and R is on the circle
Becuase F is on the center of the circle AF = 0
We know: (AF') = (AR)(AR) / (AF)
so (AF') = (1) / (0) = infinity
therefore, F' ( the inverse of F) is at infinity